This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Relation between wave equation and wave functions

I would like to know answers to these questions: Can we say that the wave equation is a predicate that wave functions (solutions to the wave equation) satisfy? Can we say that the wave equation is a ...
2
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1answer
66 views

Why wasn't Bertrand Russell surprised by the set of all sets that contain themselves?

Russell's paradox deals with the question: "Does the set of all sets that do not contain themselves, contain itself?" What about the question: "Does the set of all sets that contain themselves, ...
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27 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
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1answer
40 views

Why do metric spaces that produce the same topology have different number theoretical difficulties?

Consider finding a a point with rational distance to the corners of unit square. Under the Euclidean metric this is very hard. (unsolved) Under the "city block" or taxicab metric this is very easy ...
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1answer
29 views

Degenerate zeroes, fundamental theorem of algebra.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two ...
2
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1answer
71 views

Does PA prove that Con(PA) implies Con(ZF-I) and Con(NFU)?

I read from many sources that PA and ZF-I (a suitable axiomatization of ZF minus Infinity plus its negation) are bi-interpretable, but is PA enough to prove that they are equiconsistent? Specifically ...
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28 views

How to Prove? - If the interpretation of theory is consistent, then the interpreted theory is consistent

Let L1 and L2 be finite or recursive languages, and T a theory in the language of L2. A translation of L1 into T is an assignment to each sentence S of L1 into a sentence i(S) of L2 such that: (T0) ...
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1answer
106 views

Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
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4answers
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Are professional mathematicians concerned with formalizing infinitely many dependent choices?

I've noticed certain arguments in analysis textbooks which rely on the principle of being able to pick elements infinitely many times. For example, an argument might go "Pick $x_1\in S$ such that ...
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1answer
92 views

How do you visualize $\mathcal{P}(1)$ in constructive mathematics?

If I understand correctly, constructive mathematics doesn't prove that the powerset $\mathcal{P}(X)$ of a set $X$ is a Boolean algebra; in general, all we can say is that its a Heyting algebra. This ...
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1answer
38 views

Are there other methods of proof other than contrapositive, induction, contradiction, construction, and counter example?

I have only heard of a few methods of proof, namely, contrapositive, induction, contradiction, construction, and counter example. Are there other types of proofs?
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9answers
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Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
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0answers
48 views

What questions or areas in the foundations of mathematics remain active research fields?

By foundations of mathematics I am referring to the mathematical, logical, and philosophical foundations of the subject. I'm interested in seeing which of these have active research going on within ...
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0answers
67 views

Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? ...
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1answer
69 views

Existence of some categories in the category of categories

I'm studying category of categories. I read that when there are categories $A,B$, it is allowed to define the product $A\times B$. Equalizers and coequalizers also exist. However, there are some ...
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1answer
77 views

Has it been proved that ∨ ∧ ¬ ⇒ ⇔ ∀ ∃ and predicates are enough to express all known mathematics?

I frequently encounter this statement in math books: "All of known mathematics can be expressed in terms of elementary predicates, logical connectives, and quantifiers."
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1answer
41 views

Different Mathematics

Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he ...
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3answers
137 views

In which order should I learn the foundations of mathematics? [closed]

I know from Wikipedia that those are the four pillars of the foundations of mathematics: Proof theory Aximatic Set theory Model Theory Recursion Theory and I want to learn all of them, the problem ...
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1answer
96 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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1answer
121 views

How can mathematics work in wildly different set theories?

There are several set theories, e.g. ZFC and NF, which often have different axioms or are even outright contradictory. And yet most of other branches of mathematics, e.g. abstract algebra or topology, ...
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0answers
44 views

Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
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0answers
39 views

What type of reasoning is employed when studying metalogic?

When studying, or doing any kind of reasoning about logic, what type of logic is used? Or how is the reasoning structured?
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2answers
71 views

How are image and pre-image different from range and domain respectively?

How are image and pre-image different from range and domain respectively, in Layman's terms (as simple as possible)? Are they basically just keywords that often indicate more nuanced subsets of the ...
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2answers
387 views

Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
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0answers
26 views

Why is it impossible to prove absolute consistency of a theory falling prey to Godel's theorems?

Why is it impossible to prove absolute consistency of a theory T falling prey to Godel's theorems? I understand that a theory falling prey to Godel's second incompleteness theorem cannot prove its own ...
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1answer
18 views

Need an assistance with a specific step of a specific Division Algorithm proof

I'm trying to wrap my head around a Division Algorithm's proof. That is, Let $a, b \in \mathbb{Z}, a \neq 0$. Then there are unique $q,r \in \mathbb{Z}$ such that $b = qa + r, 0 \leq r < |a|$. ...
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2answers
815 views

Why doesn't this definition of natural numbers hold up in axiomatic set theory?

I was reading about older definitions of the natural numbers on Wikipedia here (in retrospect, not the best place to learn mathematics) and came across the following definition for the natural ...
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2answers
53 views

Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
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1answer
42 views

Why include equality in FOL for ZFC?

What are the pros and cons of working with first-order logic with equality for constructing ZFC, when all you have to do is make '$x=y$' a shorthand for: $$'\forall z [z \in x \Leftrightarrow z \in y] ...
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1answer
35 views

Any foundational theory of math falls prey to the incompleteness theorems - true or false?

I heard somewhere on the internet once something along the following lines: Any conceivable foundational theory of mathematics (be it ZFC or, if ZFC was found to be inconsistent, some modification ...
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1answer
55 views

Is isomorphism defined between large categories?

By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are ...
2
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1answer
34 views

Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of ...
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1answer
91 views

What book on Set Theory is best to understand motivation for axiomatization?

I am Master of Science in ICT, and I had always been in loved in math. On University we haven't been doing any Foundational Mathematics, the closest being Automata Theory and mention of Church-Turing ...
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3answers
4k views

Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
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0answers
46 views

Prereqisites for: Subsystems of second order arithmetic

As the title suggests, im wondering what the prerequisites for Simpsons book, Subsystems of... are? Unfortunately I cant find it in the preface. My background is a Bachelor in Philosophy and ...
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1answer
70 views

Alternatives to pure quantifier logic

Are there some alternatives for pure quantifier logic? Pure quantifier logic is axioms and rules of inference added to proposition logic to result first order logic. Are there other axioms that ...
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0answers
88 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
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2answers
344 views

Seeking a new, more natural definition of the cartesian product of sets

In "standard" set theory usually we have the definition $(a,b) := \{ \{ a \} , \{a,b\}\}$, see for example wikipedia for other similar ones. Then if we set for two sets $A,B$ $$ A\times B := \{ (a,b) ...
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3answers
77 views

Divide by 0 alternative [closed]

Cutting to the chase. I know you can't divide by zero. And I have read a good few explications for this. And I am happy with this as a fact. BUT my question is based on this: X / N = A "should ...
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1answer
84 views

Logic without content? Takeuti and Zaring

I am reading the book "Introduction to Axiomatic Set Theory" by Takeuti and Zaring, and I wonder if I understand the "language of logic" from chapter 2 properly. They write: The language of our ...
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0answers
85 views

References about positivism

To found the current set theory, it has been necessary to remove some paradoxes like the well-known Russel paradox. It was thus necessary to clarify why things like $$\{x : x \notin x\}$$ can't ...
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1answer
39 views

Constructing natural numbers as lists of units (possible infinite objects)

I'm puzzled by this question, which is more about relation between two type theoretic approaches. Nevertheless, It can be shortened to the question : When it is correct (if ever) to construct ...
4
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2answers
60 views

Axioms of Trigonometry

On Wikipedia it gives a picture of all trigonometric functions of an angle laid atop the unit circle, 1. Obviously there are other trigonometric identities, but what I'm wondering is, does ...
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1answer
80 views

Godel's proof for dummies [closed]

Can someone give me as simple-a-proof as possible for Godel's Incompleteness Theorem? I'd love to understand it more.
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3answers
162 views

Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
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0answers
19 views

Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an ...
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6answers
67 views

How can I prove that if $a^7 = b^7$ then $a=b$, with $ a,b \in \mathbb{Z} $

I've tried with divisibility, meaning that since $a$ divides $a^7$, then $a$ divides $b^7$ and in the same way b divides $a^7$, but I can't seem to go further than this. What properties of the ...
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0answers
27 views

Understanding foundational terms: notions, objects and meta-objects

I am trying to take my problem solving skill to next level. It looks like It takes a lot of mathematical discipline. Here, This post buys me to get better at proof writing. So, I think is useful to ...
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1answer
35 views

Axioms of motion (Redei version)

I try to understand the axioms of motion in Redei's "Foundationd of Euclidean and Non-Euclidean Geometries, according to F. Klein" 1968 Redei gives as Axioms: Any motion is a one to one mapping of ...
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3answers
105 views

Are the assertions “$2 + 2$ equals $4$” and “$2 +2$ is $4$” identical

Are the assertions "$2 + 2$ equals $4$" and "$2 +2$ is $4$" identical? Or is this a linguistic, psychological or murky philosophical thing rather than a mathematical thing